Delayed Differential Equation Research

[nburnett]

Homework Statement

I am researching the delayed differential equation x'(t) = x(t/2) and haven't made much progress. Does anyone have suggestions?

Homework Equations

I have proved that for any initial condition, a solution exists, and it is unique, but the proofs are fairly long, so I won't post them here. Both of the proofs rely on the equation satisfying a locally Lipschitz condition.

The Attempt at a Solution

The only progress I have made on finding a solution is the easy case where x(0) = 0, in which case x = 0 satisfies the equation.

 

[lippyka]

I have considered various methods, such as Picard iteration and numerical approximations, but have not had much success. It seems like the equation is too nonlinear to be amenable to these methods.I have also considered Laplace transforms, but I'm not sure how to apply them in this case.One idea I had was to use a series expansion of x(t) around t = 0; however, this approach seems difficult to implement in practice.I'm at a loss for how to proceed, so any advice or suggestions would be greatly appreciated.